{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:V4I4H76473S3S2HKF2QUHUEROL","short_pith_number":"pith:V4I4H764","canonical_record":{"source":{"id":"2509.08599","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-09-10T13:57:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"995bf93a82f5075741131a35433662724eb91982a5a165021d31ad957a63bd44","abstract_canon_sha256":"9074aca30e9c7f0f4d882a9f261f78920beab6026a40ed14b707ea7d4f72cada"},"schema_version":"1.0"},"canonical_sha256":"af11c3ffdcfee5b968ea2ea143d09172e61640400745fddbe10f96a2b2e0eb15","source":{"kind":"arxiv","id":"2509.08599","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08599","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08599v1","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08599","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_12","alias_value":"V4I4H76473S3","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_16","alias_value":"V4I4H76473S3S2HK","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_8","alias_value":"V4I4H764","created_at":"2026-07-05T12:08:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:V4I4H76473S3S2HKF2QUHUEROL","target":"record","payload":{"canonical_record":{"source":{"id":"2509.08599","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-09-10T13:57:55Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"995bf93a82f5075741131a35433662724eb91982a5a165021d31ad957a63bd44","abstract_canon_sha256":"9074aca30e9c7f0f4d882a9f261f78920beab6026a40ed14b707ea7d4f72cada"},"schema_version":"1.0"},"canonical_sha256":"af11c3ffdcfee5b968ea2ea143d09172e61640400745fddbe10f96a2b2e0eb15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:08:32.852633Z","signature_b64":"SPisIOqTLqx8n3tMVfLAn40uItnOuK+dyAKKtjvrcnP91t8mtp1JXZVg7psXjI4VSwbYEoP/sFjXFLnRGQb8DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af11c3ffdcfee5b968ea2ea143d09172e61640400745fddbe10f96a2b2e0eb15","last_reissued_at":"2026-07-05T12:08:32.852071Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:08:32.852071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.08599","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:08:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VllqFiBkUGGbEs5hdJvR3cA+CrvNw8vaLX7741I3ayzWeHGpq2wsx79aQzfs50L8wZ62MensVeLKDbpYQycHAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T07:39:09.985555Z"},"content_sha256":"34bd86608f285996fbdbc18d13742d4528f6c0ec5aade3af07cb217d20d960dd","schema_version":"1.0","event_id":"sha256:34bd86608f285996fbdbc18d13742d4528f6c0ec5aade3af07cb217d20d960dd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:V4I4H76473S3S2HKF2QUHUEROL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Diophantine Frobenius Problem revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hao Zhang, Weijia Wang, Yuchen Ding","submitted_at":"2025-09-10T13:57:55Z","abstract_excerpt":"Let $k\\ge 2$ and $a_1, a_2, \\cdots, a_k$ be positive integers with \\[ \\gcd(a_1, a_2, \\cdots, a_k)=1. \\] It is proved that there exists a positive integer $G_{a_1, a_2, \\cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \\[ n=a_1x_1+a_2x_2+\\cdots+a_kx_k, \\quad (x_1, x_2, \\cdots, x_k\\in\\mathbb{Z}_{\\ge 0},~\\gcd(x_1, x_2, \\cdots, x_k)=1). \\] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08599","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.08599/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:08:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dvMRHoV6rF++RkQYnH/cj4mPgXDe0y9sini16PyhVS9yP4XP6/hcgc9yMWbRkVf2WsIAPxfWqs6u5okp7bMRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T07:39:09.986494Z"},"content_sha256":"cf613215e5b35dc4ed84694547b59fd17ce47c388e1bb929aad2a4639c87450d","schema_version":"1.0","event_id":"sha256:cf613215e5b35dc4ed84694547b59fd17ce47c388e1bb929aad2a4639c87450d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4I4H76473S3S2HKF2QUHUEROL/bundle.json","state_url":"https://pith.science/pith/V4I4H76473S3S2HKF2QUHUEROL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4I4H76473S3S2HKF2QUHUEROL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-10T07:39:09Z","links":{"resolver":"https://pith.science/pith/V4I4H76473S3S2HKF2QUHUEROL","bundle":"https://pith.science/pith/V4I4H76473S3S2HKF2QUHUEROL/bundle.json","state":"https://pith.science/pith/V4I4H76473S3S2HKF2QUHUEROL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4I4H76473S3S2HKF2QUHUEROL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:V4I4H76473S3S2HKF2QUHUEROL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9074aca30e9c7f0f4d882a9f261f78920beab6026a40ed14b707ea7d4f72cada","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-09-10T13:57:55Z","title_canon_sha256":"995bf93a82f5075741131a35433662724eb91982a5a165021d31ad957a63bd44"},"schema_version":"1.0","source":{"id":"2509.08599","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.08599","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"arxiv_version","alias_value":"2509.08599v1","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.08599","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_12","alias_value":"V4I4H76473S3","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_16","alias_value":"V4I4H76473S3S2HK","created_at":"2026-07-05T12:08:32Z"},{"alias_kind":"pith_short_8","alias_value":"V4I4H764","created_at":"2026-07-05T12:08:32Z"}],"graph_snapshots":[{"event_id":"sha256:cf613215e5b35dc4ed84694547b59fd17ce47c388e1bb929aad2a4639c87450d","target":"graph","created_at":"2026-07-05T12:08:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2509.08599/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $k\\ge 2$ and $a_1, a_2, \\cdots, a_k$ be positive integers with \\[ \\gcd(a_1, a_2, \\cdots, a_k)=1. \\] It is proved that there exists a positive integer $G_{a_1, a_2, \\cdots, a_k}$ such that every integer $n$ strictly greater than it can be represented as the form \\[ n=a_1x_1+a_2x_2+\\cdots+a_kx_k, \\quad (x_1, x_2, \\cdots, x_k\\in\\mathbb{Z}_{\\ge 0},~\\gcd(x_1, x_2, \\cdots, x_k)=1). \\] We then investigate the size of $G_{a_1, a_2}$ explicitly. Our result strengthens the primality requirement of $x$'s in the classical Diophantine Frobenius Problem.","authors_text":"Hao Zhang, Weijia Wang, Yuchen Ding","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-09-10T13:57:55Z","title":"The Diophantine Frobenius Problem revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08599","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:34bd86608f285996fbdbc18d13742d4528f6c0ec5aade3af07cb217d20d960dd","target":"record","created_at":"2026-07-05T12:08:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9074aca30e9c7f0f4d882a9f261f78920beab6026a40ed14b707ea7d4f72cada","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-09-10T13:57:55Z","title_canon_sha256":"995bf93a82f5075741131a35433662724eb91982a5a165021d31ad957a63bd44"},"schema_version":"1.0","source":{"id":"2509.08599","kind":"arxiv","version":1}},"canonical_sha256":"af11c3ffdcfee5b968ea2ea143d09172e61640400745fddbe10f96a2b2e0eb15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af11c3ffdcfee5b968ea2ea143d09172e61640400745fddbe10f96a2b2e0eb15","first_computed_at":"2026-07-05T12:08:32.852071Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:08:32.852071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SPisIOqTLqx8n3tMVfLAn40uItnOuK+dyAKKtjvrcnP91t8mtp1JXZVg7psXjI4VSwbYEoP/sFjXFLnRGQb8DQ==","signature_status":"signed_v1","signed_at":"2026-07-05T12:08:32.852633Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.08599","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:34bd86608f285996fbdbc18d13742d4528f6c0ec5aade3af07cb217d20d960dd","sha256:cf613215e5b35dc4ed84694547b59fd17ce47c388e1bb929aad2a4639c87450d"],"state_sha256":"9d85b3b0016ae93ffa48cbff240d1ee351a5174c3fbe19af4423b608b06e2fcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8LKutqLsVud69PFwcEbcCy3kJHqcQaOuKwsvr/tP52cGK3OLScsAuom6fULsrkjrvF+YmADf8+/hxdDwEWITBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T07:39:09.995167Z","bundle_sha256":"05889588bd203fa9a487af5d10a5519333f777c90f2a7a4ee19d0ff799811eff"}}